295:
theoretic terms, one speaks of probability, noise, and divergence. A stochastic system is probabilistic, but may still very well be a continuous stochastic system, that is, have an information dimension of one, and ultimately represent the same amount of information as a chaotic system. Indeed, one can have a stochastic chaotic system, with an arbitrary rate infromation decay/divergence. Chaotic and stochastic are orthogonal classifications. A linear(non-chaotic) and non-stochastic system simply evades the question of information and therefore predictability altogether. We must therefore disclude it from any concept of predictability or randomness. Indeed, randomness already discludes all non-stochastic systems. Where a system is "random", it has "divergence", times a scalar. A system may have divergence distributed evenly in resepct to a real-valued parameter. In this case, one simply adds a "noise" term: + N. This is so-called "randomness". In a chaotic system, on the other hand, the divergence is concentrated by a point. Regardless, one has convergence and divergence, and in either case noise is noise is noise. i.e. randomness is divergence is noise. In terms of "causuality" in the markov sense, there is really no fundamental discriminatory factor. It still remains possible, ofcourse, for one to make a (superficial) mathematical distinction, but one should not look for any ontological distinction - it is ontologically unnecessary for there to be a distinction.
310:, inextricably, noise, and thus noise, since it cannot be reduced, replaced, or removed, must ultimatly be in-itself, regardless of whether it is "extrinsic" or "intrinsic". (The question was never what name we give to the difference of the being-of-noise, but rather the simply differential character of the difference itself.) The more fundamental question is not intrinisic vs. extrinsic, but in-itself (essential) vs. phenomenological. But the two positions are informationally equaivalent/indistinguishable: one nonetheless has "noise", "information", and everything else. There is no way to statistically distingush among the consequences beyond a mere quantification of the "noise rate". --
679:
687:
74:
53:
741:
84:
150:
22:
302:
A chaotic is neccessarily a non-equilibrium system; chaotic systems only exist where there is a flow of energy, and thus also neccessarily, a flow of information. The "unpredictability" of a chaotic system is part of this flow, just as the "noise" in a stochastic system is an influx of information.
284:, operates on un-computable numbers. This means that even a theoretically ("in principle") "perfect" computer could not compute the vast majority of trajectories. And by vast majority, I mean 99.999999999...%. For a formal definition of computability, see Turing's essay "On Computable Numbers".
1074:
These two graphs have to be some of the worst I've ever seen. They are so badly labelled, and give no idea to a reasonable user as to what the they are showing (the right graph specifically). "Parameter" should at least be labelled "parameter, r", and none of t, T, f and k are anywhere defined
294:
I was going to fix this paragraph myself, and I had much trouble, because when ever I encountered a "not the same as..." I thought for a minute and eventually answered "Yes it is." The proper way to speak of predictability/unpredictability is in information-theoretic terms. In information
773:
The first graph on the page (the animation) has mislabeled axes. The horizontal axis should be "n" to be consistent with the equation given in the article. Took me forever to figure out what it was talking about. I'd do it myself but I don't know how to make animated gifs.
1075:
properly. And " t=nT" is not how you properly label an axis. First, either it is "t" or it is "n" (and not "" either), and secondly, some description would be nice. I've seen better graphs than this on vixra. It needs to be sorted out or removed completely, unfortunately.
252:
Yes. If you iterate practically any function that's not linear (not a straight line), then there will be some starting values that give chaotic (although, of course, perfectly deterministic) results. The FractInt software comes with lots of them. -- DavidCary
600:
There are methods for proving that period 3 begins at exactly r = 1 + Sqrt(8). I can provide a published proof if necessary, but also did the proof myself as a presentation in a chaos theory class. I think this section could benefit by mentioning this.
298:
But I have not yet been complete enough, one might still raise the objection that a chaotic system is fully specified by a finite set of symbols and is non-stochastic. However, my point is that this is completely irrelevant. The symbols are arbitrary.
569:
306:
But there is also the philosophical problem of "noise" as extrinsic "unknown" (determinism) - the scientific deterministic assumption - vs. noise as an in-itself (free-will). But in either case there
326:=1). For example if r was 4, you wouldn't get much chaotic behavior if you started with .75 .Is the talk about period and chaotic behavior talking about a general seed such as a transcendental one?
753:
tracing the first few iterates in order to give a visual explanation of the behavior of the chaotic "shadows". Not sure where/if it's worth publishing it... so I just mention it here, for now.
408:
487:
605:
07:41, 23 May 2007 (UTC) Please give a reference to published proof in article and if I'm not wrong good place for a your proof is wikibooks ( with link to wikibooks in article). --
249:
I.e., you have the
Mandelbrot set and the logistic map being a degree 2 polynomial. Are there any interesting examples with degree 3? Degree 4? Other types of functions?
624:
It would be interesting to read about connections to biology. Are there interesting observations in nature predicted by this? (Most interesting seems the range 3-3.45)
191:
I can't follow the algebra here. Mandelbrot is z = z^2 + c and logistic is x = r x (1-x). I don't think the formula given above converts these formulas into each other.
795:
The right limit for the t axis is too big. I suggest making it (0,75) instead of (0,200). It's really too difficult to see the transient dynamics in the current GIF.
1009:
1005:
991:
893:
The population goes negative, which doesn't really make sense as a population model? I imagine that's why nobody seems to have bothered to consider this case.
1108:
1103:
158:
130:
723:
I could upload the
Mathematica .nb file. The algorithm is similar to the one described in the old image. Where in commons would I put the info? --
140:
750:
1069:
815:
1113:
631:
777:
653:
XMIN : 0 XMAX : 1 XSCL : .1 YMIN : 0 YMAX : 1 YSCL : .1 XINIT : $ A2/.5+ WEBSTART : 0 WEBSTOP : 50 CURRENT : 1'-'*$ B1+2** REFERENCE : '
967:
496:
1098:
987:
When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.
318:"With r between 3 and 1+√6 (approximately 3.45)..." It seems like there's a typo in the 1+6, but I'm not sure what it should be --
957:
303:
The difference is simply whether the distribution of this influx - this flow - is flat with respect to a given measurement.
97:
58:
1052:
33:
675:
I created a higher resolution orbit diagram for the logistic map and was wondering if indeed it is worth putting here.
266:
One of the quantities X sub n on the right side of the equation should be an X sub 0. I'm not sure which one, though.
586:
345:
325:
Why does it keep saying independent of inital seed? It seems like (r-1)/r is a solution no matter what r is (r: -->
635:
583:
Yes, that's correct. There are also explicit solutions for r = 2 and r = -2 More information can be found here:
327:
1008:
to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the
417:
678:
781:
1043:
977:
949:
843:
800:
714:
610:
945:
920:
898:
728:
699:
89:
1027:
If you have discovered URLs which were erroneously considered dead by the bot, you can report them with
1015:
39:
948:. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit
602:
968:
https://web.archive.org/web/20090821122553/http://www.users.bigpond.com/pmurray/Java/LogisticMap.html
627:
311:
288:
21:
686:
883:
857:
839:
823:
796:
710:
606:
1012:
before doing mass systematic removals. This message is updated dynamically through the template
1028:
916:
894:
758:
319:
233:
971:
861:
827:
1035:
958:
https://web.archive.org/web/20060211001015/http://www.ibiblio.org/e-notes/MSet/Logistic.htm
740:
724:
695:
661:
106:
73:
52:
1080:
1068:
994:, "External links modified" talk page sections are no longer generated or monitored by
814:
192:
183:
1034:
If you found an error with any archives or the URLs themselves, you can fix them with
1092:
879:
572:
908:
941:
754:
229:
149:
1001:
584:
277:
961:
856:
Thanks Matt, glad you like it. I scaled it down a bit and put it in. - Regards
1000:. No special action is required regarding these talk page notices, other than
657:
79:
1076:
878:
Should there be description on what happens if the value of r is below 0? --
978:
https://web.archive.org/web/20070415033246/http://lectures.nsitlounge.in/
102:
1084:
1057:
924:
902:
887:
865:
847:
831:
804:
785:
762:
732:
718:
703:
665:
639:
614:
589:
575:
330:
237:
564:{\displaystyle x_{n}={\frac {1}{2}}-{\frac {1}{2}}\cos(2^{n}\phi )}
739:
685:
677:
280:...". This statement is logically false, because a logistic map,
981:
709:
could you put the code ( in commons) and describe algorithm ? --
15:
148:
972:
http://www.users.bigpond.com/pmurray/Java/LogisticMap.html
952:
for additional information. I made the following changes:
838:
Much better. I love it! Can you add it to the main page
596:
Chaos (or period 3) begins at precisely r = 1 + Sqrt(8)
340:
if r = 4, the logistic map appears to be solvable: set
246:
Are there any other formulae that have this property.
499:
420:
348:
1004:using the archive tool instructions below. Editors
563:
481:
402:
260:There seems to be a problem with this equation:
423:
351:
270:Nope. Both on the right are supposed to be x_n.
990:This message was posted before February 2018.
403:{\displaystyle \,\!\phi =\cos ^{-1}(1-2x_{0})}
256:(Should this answer go on the article page ?)
650:There is another way, using WEBPLOT program:
8:
962:http://ibiblio.org/e-notes/MSet/Logistic.htm
694:Please let me know what you think. Thanks!
101:, which collaborates on articles related to
915:= −2, which also admits an exact solution.
482:{\displaystyle \,\!x_{n+1}=4x_{n}(1-x_{n})}
47:
940:I have just modified 3 external links on
549:
526:
513:
504:
498:
470:
451:
429:
419:
391:
363:
347:
212:-1/2)) and use the change of variables: z
907:I was wrong about this: MathWorld has a
751:Image:LogisticMap BifurcationDiagram.png
276:Regarding "also given access to perfect
421:
349:
49:
19:
7:
157:This article is within the field of
95:This article is within the scope of
182:The logistic map is related to the
38:It is of interest to the following
14:
944:. Please take a moment to review
1109:Systems articles in chaos theory
1104:High-importance Systems articles
1067:
813:
82:
72:
51:
20:
135:This article has been rated as
982:http://lectures.nsitlounge.in/
866:05:15, 15 September 2014 (UTC)
733:06:28, 15 September 2011 (UTC)
719:19:12, 14 September 2011 (UTC)
704:08:39, 12 September 2011 (UTC)
558:
542:
476:
457:
397:
375:
186:by the equation c=(1-(r-1))/4.
1:
1085:20:58, 8 September 2019 (UTC)
848:15:42, 8 September 2014 (UTC)
682:High Resolution Orbit Diagram
576:20:55, 18 December 2006 (UTC)
571:which can be easily checked.
204:Rewrite the logistic map as x
115:Knowledge:WikiProject Systems
1114:WikiProject Systems articles
888:13:58, 20 October 2014 (UTC)
763:18:02, 10 October 2018 (UTC)
590:16:29, 10 January 2007 (UTC)
263:x_{n+1} = r x_n (1 - x_n),
118:Template:WikiProject Systems
1058:06:55, 5 January 2018 (UTC)
925:12:09, 31 August 2018 (UTC)
671:Higher Resolution Orbit Map
615:14:37, 13 August 2009 (UTC)
238:21:55, 12 August 2009 (UTC)
1130:
1021:(last update: 5 June 2024)
937:Hello fellow Wikipedians,
903:10:09, 27 March 2015 (UTC)
666:01:24, 23 April 2010 (UTC)
228:+ c with c=(1-(r-1))/4.
141:project's importance scale
646:Logistic map with WEBPLOT
640:02:55, 7 April 2008 (UTC)
314:06:21, 28 Feb 2004 (UTC)
291:20:11, 26 Feb 2004 (UTC)
195:02:24 Sep 30, 2002 (UTC)
156:
134:
67:
46:
1099:B-Class Systems articles
832:08:27, 9 July 2014 (UTC)
805:22:42, 22 May 2014 (UTC)
786:06:56, 19 May 2014 (UTC)
690:Orbit Diagram in Article
331:00:41, 23 May 2006 (UTC)
933:External links modified
745:
691:
683:
565:
483:
404:
328:Hiiiiiiiiiiiiiiiiiiiii
153:
90:Systems science portal
28:This article is rated
1063:Badly labelled graphs
743:
689:
681:
587:Scribblesinmindscapes
566:
484:
405:
152:
1002:regular verification
769:Inconsistent diagram
497:
418:
346:
176:Very old discussions
992:After February 2018
98:WikiProject Systems
1046:InternetArchiveBot
997:InternetArchiveBot
746:
692:
684:
561:
479:
424:
422:
400:
352:
350:
154:
34:content assessment
1022:
749:I scribbled over
642:
630:comment added by
534:
521:
173:
172:
169:
168:
165:
164:
1121:
1071:
1056:
1047:
1020:
1019:
998:
874:r values below 0
817:
810:How about this?
625:
570:
568:
567:
562:
554:
553:
535:
527:
522:
514:
509:
508:
488:
486:
485:
480:
475:
474:
456:
455:
440:
439:
409:
407:
406:
401:
396:
395:
371:
370:
123:
122:
121:Systems articles
119:
116:
113:
92:
87:
86:
85:
76:
69:
68:
63:
55:
48:
31:
25:
24:
16:
1129:
1128:
1124:
1123:
1122:
1120:
1119:
1118:
1089:
1088:
1065:
1050:
1045:
1013:
1006:have permission
996:
950:this simple FaQ
935:
876:
793:
771:
673:
654:
648:
632:130.242.107.211
622:
598:
545:
500:
495:
494:
466:
447:
425:
416:
415:
387:
359:
344:
343:
338:
264:
227:
223:
219:
215:
211:
207:
178:
137:High-importance
120:
117:
114:
111:
110:
107:systems science
88:
83:
81:
62:High‑importance
61:
32:on Knowledge's
29:
12:
11:
5:
1127:
1125:
1117:
1116:
1111:
1106:
1101:
1091:
1090:
1064:
1061:
1040:
1039:
1032:
985:
984:
976:Added archive
974:
966:Added archive
964:
956:Added archive
934:
931:
930:
929:
928:
927:
875:
872:
871:
870:
869:
868:
851:
850:
835:
834:
820:
819:
818:
792:
789:
778:205.175.97.169
770:
767:
766:
765:
747:
744:Early iterates
736:
735:
721:
672:
669:
652:
647:
644:
621:
618:
597:
594:
593:
580:
560:
557:
552:
548:
544:
541:
538:
533:
530:
525:
520:
517:
512:
507:
503:
478:
473:
469:
465:
462:
459:
454:
450:
446:
443:
438:
435:
432:
428:
399:
394:
390:
386:
383:
380:
377:
374:
369:
366:
362:
358:
355:
337:
334:
323:
316:
274:
262:
258:
244:
242:
225:
221:
217:
213:
209:
208:= r ( 1/4 - (x
205:
202:
201:
200:
189:
188:
184:Mandelbrot set
177:
174:
171:
170:
167:
166:
163:
162:
155:
145:
144:
133:
127:
126:
124:
94:
93:
77:
65:
64:
56:
44:
43:
37:
26:
13:
10:
9:
6:
4:
3:
2:
1126:
1115:
1112:
1110:
1107:
1105:
1102:
1100:
1097:
1096:
1094:
1087:
1086:
1082:
1078:
1072:
1070:
1062:
1060:
1059:
1054:
1049:
1048:
1037:
1033:
1030:
1026:
1025:
1024:
1017:
1011:
1007:
1003:
999:
993:
988:
983:
979:
975:
973:
969:
965:
963:
959:
955:
954:
953:
951:
947:
943:
938:
932:
926:
922:
918:
914:
910:
906:
905:
904:
900:
896:
892:
891:
890:
889:
885:
881:
873:
867:
863:
859:
855:
854:
853:
852:
849:
845:
841:
840:MATThematical
837:
836:
833:
829:
825:
821:
816:
812:
811:
809:
808:
807:
806:
802:
798:
797:MATThematical
790:
788:
787:
783:
779:
775:
768:
764:
760:
756:
752:
748:
742:
738:
737:
734:
730:
726:
722:
720:
716:
712:
711:Adam majewski
708:
707:
706:
705:
701:
697:
688:
680:
676:
670:
668:
667:
663:
659:
651:
645:
643:
641:
637:
633:
629:
619:
617:
616:
612:
608:
607:Adam majewski
604:
595:
592:
591:
588:
585:
581:
578:
577:
574:
555:
550:
546:
539:
536:
531:
528:
523:
518:
515:
510:
505:
501:
492:
489:
471:
467:
463:
460:
452:
448:
444:
441:
436:
433:
430:
426:
413:
410:
392:
388:
384:
381:
378:
372:
367:
364:
360:
356:
353:
341:
335:
333:
332:
329:
322:
321:
315:
313:
309:
304:
300:
296:
292:
290:
285:
283:
279:
273:
272:-- DavidCary
271:
267:
261:
257:
254:
250:
247:
243:
240:
239:
235:
231:
198:
197:
196:
194:
187:
185:
180:
179:
175:
160:
151:
147:
146:
142:
138:
132:
129:
128:
125:
108:
104:
100:
99:
91:
80:
78:
75:
71:
70:
66:
60:
57:
54:
50:
45:
41:
35:
27:
23:
18:
17:
1073:
1066:
1044:
1041:
1016:source check
995:
989:
986:
942:Logistic map
939:
936:
917:Double sharp
912:
911:considering
895:Double sharp
877:
794:
776:
772:
693:
674:
655:
649:
623:
603:68.55.58.255
599:
582:
579:
493:
490:
414:
411:
342:
339:
324:
320:Kevinatilusa
317:
307:
305:
301:
297:
293:
286:
282:in principle
281:
275:
269:
268:
265:
259:
255:
251:
248:
245:
241:
220:). Result: z
203:
190:
181:
159:Chaos theory
136:
96:
40:WikiProjects
626:—Preceding
278:computation
216:= r(1/2 - x
1093:Categories
1053:Report bug
725:Efecretion
696:Efecretion
312:Kevin Baas
289:Kevin Baas
1036:this tool
1029:this tool
822:Regards,
791:GIF, axes
193:AxelBoldt
1042:Cheers.—
880:Artman40
628:unsigned
573:Scythe33
491:we have
412:then if
199:I agree.
946:my edit
620:Biology
230:Abelian
139:on the
112:Systems
103:systems
59:Systems
30:B-class
858:Snaily
824:Snaily
36:scale.
658:zzo38
1081:talk
1077:Krea
921:talk
909:page
899:talk
884:talk
862:talk
844:talk
828:talk
801:talk
782:talk
759:talk
729:talk
715:talk
700:talk
636:talk
611:talk
336:note
234:talk
131:High
105:and
1010:RfC
980:to
970:to
960:to
755:ale
537:cos
361:cos
287:--
224:= z
222:n+1
206:n+1
1095::
1083:)
1023:.
1018:}}
1014:{{
923:)
901:)
886:)
864:)
846:)
830:)
803:)
784:)
761:)
731:)
717:)
702:)
664:)
656:--
638:)
613:)
556:ϕ
540:
524:−
464:−
382:−
373:
365:−
354:ϕ
308:is
236:)
1079:(
1055:)
1051:(
1038:.
1031:.
919:(
913:r
897:(
882:(
860:(
842:(
826:(
799:(
780:(
757:(
727:(
713:(
698:(
662:✉
660:(
634:(
609:(
559:)
551:n
547:2
543:(
532:2
529:1
519:2
516:1
511:=
506:n
502:x
477:)
472:n
468:x
461:1
458:(
453:n
449:x
445:4
442:=
437:1
434:+
431:n
427:x
398:)
393:0
389:x
385:2
379:1
376:(
368:1
357:=
232:(
226:n
218:n
214:n
210:n
161:.
143:.
109:.
42::
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
